50
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 1, FEBRUARY 2006
An Improved Flux Observer Based on PLL
Frequency Estimator for Sensorless Vector
Control of Induction Motors
Mihai Comanescu, Student Member, IEEE, and Longya Xu, Fellow, IEEE
Abstract—This paper presents an improved method of flux
estimation for sensorless vector control of induction motors based
on a phase locked loop (PLL) programmable low-pass filter (LPF)
and a vector rotator. A PLL synchronized with the voltage vector is
used for stator frequency estimation. The pure integration of the
stator voltage equations is difficult to implement and LPFs with
a fixed cutoff provide good estimates only in the relatively high
frequency range—at low frequencies, the estimates fail in both
magnitude and phase. The method proposed corrects the above
problem for a wide range of speeds. Simulations and experimental
results on a 0.25-hp three-phase induction machine verify the
validity of the approach.
Index Terms—Induction machine, phase locked loop (PLL),
sensorless control, vector control.
N OMENCLATURE
Vs = [Vα , Vβ ]
Is = [Iα , Iβ ]
es = [eα , eβ ]
λs = [λαs , λβs ]
λr = [λαr , λβr ]
Vd , Vq
Id , Iq
Van , Vbn
ωs
ωr
np
Rs
Lm
Ls , Lr
Tr
ωc
λ1s
k
Stator voltage in stationary reference
frame.
Currents in stationary reference frame.
Back EMFs in stationary reference frame.
Stator fluxes in stationary reference
frame.
Rotor fluxes in stationary reference
frame.
Voltages in synchronous reference frame.
Currents in synchronous reference frame.
Line-neutral voltages.
Synchronous frequency.
Rotor mechanical speed.
Number of pole pairs.
Stator resistance.
Magnetizing inductance.
Stator and rotor total inductances.
Rotor time constant.
LPF cutoff frequency.
Intermediate stator flux vector.
Ratio of cutoff frequency to stator
frequency.
Manuscript received April 21, 2004; revised September 10, 2004. Abstract
published on the Internet November 25, 2005.
M. Comanescu was with the Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43210 USA. He is now with
Azure Dynamics, Woburn, MA 01801-2103 USA.
L. Xu is with the Department of Electrical and Computer Engineering, The
Ohio State University, Columbus, OH 43210 USA (e-mail: xu.12@osu.edu).
Digital Object Identifier 10.1109/TIE.2005.862317
dc
edc
α , eβ
θv
θr
Ta , Tb , Tc
σ = 1 − L2m /Ls Lr
dc offsets of α, β back EMFs.
Angle of stator voltage vector.
Angle of rotor flux vector.
Duty cycles of the inverter switches.
Leakage factor.
I. I NTRODUCTION
E
STIMATION of the fluxes using the voltage model (VM)
observer is a convenient method for the sensorless vector
control of induction motor drives. Techniques based on rotor
equations require knowledge of the mechanical speed and depend heavily on the accuracy of the rotor time constant. Full
order observers are more difficult to implement and rely on an
even larger set of machine parameters [1]–[6].
The VM observer is both attractive and important for several
reasons. First, if necessary, the critical parameter (Rs ) can be
estimated in real time by using stator-mounted temperature sensors [7], and accurate stator fluxes can be obtained. Second, the
coefficients in the equations linking the stator and rotor fluxes
do not vary significantly with operating conditions [8]; thus, a
good estimation of the rotor fluxes is possible and the decoupled
rotor direct field orientation (DFO) method can be used. Third,
dual-flux model reference adaptive system (MRAS) observers
use the VM as a reference in order to estimate the motor speed.
Finally, the estimate of the flux angle (in either stator or rotor
DFO method) is crucial for the stability and the performance of
a sensorless drive.
Many researchers have addressed issues related to flux observers and the main practical difficulty is well known—the
offsets and drifts present in the motor back electromotive forces
(EMFs) make the pure integration very difficult. To solve this
problem, most methods described in the literature avoid the
pure integration and are based on low-pass filters (LPFs) with
either a fixed or a variable cutoff frequency [1]–[8]. An offset
compensation method and a pure integration have been reported
in [9], [10]. Excellent experimental results are reported at a low
frequency; however, the complexity, computational burden, and
accuracy issues almost prohibit its implementation on a lowcost fixed-point processor.
The synthesis of the stator fluxes using a fixed-cutoff LPF
is easy to implement. Selection of an appropriate cutoff frequency can be done by experimental trials (usual values are
2–4 Hz); generally, this value depends on the hardware setup,
quality of voltage, current feedback signals, A–D conversion
precision, system noise, etc. With the above LPF replacing the
0278-0046/$20.00 © 2006 IEEE
COMANESCU AND XU: AN IMPROVED FLUX OBSERVER FOR SENSORLESS VECTOR CONTROL OF INDUCTION MOTORS
pure integrator, the dc offset problem can be largely alleviated
and the flux estimates are close to the real ones if the stator
frequency is five times or higher than the filter cutoff. However,
the magnitude and angle errors are significant in the lower
frequency range; this is briefly reviewed in Section II.
Three new integration algorithms are presented in [11],
where fixed cutoff filters and feedback are used to emulate
a pure integration. The method most suited for ac machine
flux estimation exploits the orthogonality of the back EMF and
stator flux vectors.
Programmable LPFs are used in [1]–[3] and [7] for the stator
flux estimation. The αβ-axis back EMFs are fed into variablecutoff filters, and the outputs are corrected by a vector rotator.
Filter constants and vector-rotator parameters are functions of
the stator frequency (which must be estimated). Therefore,
the accuracy of the frequency estimate is a key factor in the
overall flux-estimation method. In [1]–[3], the stator frequency
is computed based on the induction-machine equations.
In this paper, this method is proposed—the stator fluxes are
synthesized using programmable LPFs and a vector rotator
similar to [1]–[3]. The novelty is that, for this type of application, the stator frequency is estimated with a phase locked
loop (PLL). The PLL synthesizes a rotating reference frame
that always tries to instantaneously align with the machine’s
voltage vector. The stator frequency is a stable by-product of the
PLL process. The idea of using a PLL for frequency estimation
comes from uninterruptible power supply (UPS) systems and
was inspired by [12].
The PLL only uses the voltage vector and is likely to yield
a better frequency estimate under real conditions (imperfections, offsets, etc.) than the method based on the machine’s
equations.
Simulations and experimental results show that, with proper
tuning, the PLL can estimate the stator frequency with a sufficient bandwidth. Overall, good flux estimates are obtained and
the method can be used for sensorless field orientation control
of induction motors with improved performance.
51
Fig. 1. Block diagram of the programmable LPF and the vector rotator.
Fig. 2. Position of vector λ1s as a function of the design parameter k.
integration are
ωs
+ ωc2
(4)
ωs
π
− tan−1 .
2
ωc
(5)
εmag = εphase =
ωs2
II. VM O BSERVER W ITH F IXED -C UTOFF LPF
At low values of ωs , the estimator yields a flux vector whose
magnitude is smaller than real and whose phase lags to the back
EMF by an angle less than 90◦ . As ωs increases, εmag tends to 1
while εphase tends to 0 and the estimate is close to the real flux.
Additional details regarding the traditional VM observer and
the errors occurring when integrators are substituted by LPFs
can be found in [1]–[4].
The VM observer estimates the stator and rotor fluxes according to
III. VM O BSERVER W ITH P ROGRAMMABLE LPF
λs =
λr =
(Vs − Rs Is )dt =
Lr
(λs − Ls σIs ).
Lm
es dt
(1)
(2)
With the pure integrator replaced by an LPF with the cutoff
frequency ωc , the relationship between the stator flux and back
EMF becomes
λs
1
=
.
es
s + ωc
(3)
The magnitude and phase errors between the flux vector
obtained by (3) and the flux vector that would result from pure
Fig. 1 shows the block diagram of the proposed programmable LPF and vector rotator.
The αβ components of the back-EMF vector are fed into
the two LPFs whose cutoff frequency is k times the estimated
stator frequency. The output of the LPFs (λ1s ) is compensated
in magnitude and phase to obtain the flux vector that would
result from an analytical integration of (1). The overall structure
of the estimator is designed to emulate the frequency function
of a pure integrator and to avoid output saturation. This is
basically similar to the methods in [1]–[3]. Fig. 2 shows a
vector diagram and the position of λ1s as a function of k. For
1
◦
k = 1, the intermediate
√ vector λs lags the back EMF by 45
and its magnitude is 2 times smaller than that of the vector
λs . As k increases, λ1s lags the back EMF by an angle less
than 45◦ and its magnitude decreases. In implementation, the
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 1, FEBRUARY 2006
αβ components of λ1s are available at every sampling time. For
ωs > 0, the pair of stator fluxes λαs , λβs is constructed as
1 + k 2 λ1αs cos ϕ + λ1βs sin ϕ
λβs = 1 + k 2 −λ1αs sin ϕ + λ1βs cos ϕ
1
π
ϕ = − tan−1
.
2
k
λαs =
(6)
(7)
(8)
Since k is a constant, the compensation gain and the sine and
cosine of the compensation angle can be computed off-line. The
performance of this estimator depends mainly on the accuracy
of the ωs estimate but also on the choice of k. Thus, at high values of k, the cutoff frequency of the LPF is high, and this helps
attenuate the offsets present in the back EMFs. However, the
compensation gain is also high and the vector rotator reamplifies the offsets. A steady-state offset analysis can be performed
to estimate the worst case offsets that appear in the flux components λαs , λβs . If the offsets at the LPF inputs are denoted as
dc
edc
α and eβ , the maximum offsets at the compensator output are
dc
λdc
α,max = λβ,max =
√
1 dc dc 1 + k2
eα + eβ .
·
k
|ωs |
(9)
The expression above shows that output offsets are frequency
dependent and the behavior may worsen if ωs is very small.
Comparison to the offsets yielded by a fixed-cutoff LPF, assumdc
ing k = 1 and |edc
α | = |eβ |, shows
√ that the programmable LPF
is better for frequencies ωs ≥ 2 2ωc . An additional feature of
the estimator is the existence of an optimal k; k = 1 minimizes
the first factor in (9).
The biggest disadvantage of this method is the requirement
of the vector rotator to know the sign of the stator frequency.
For low and especially near zero frequency, ωs may temporarily
change sign or oscillate around zero. The uncertainty in the sign
of ωs seriously upsets the angle compensation.
Fig. 3.
Proposed PLL for frequency estimation in IM drives.
frequencies must be achieved. The block diagram of the PLL
proposed in this paper is shown in Fig. 3.
For the PLL shown above, the stator voltage is used as a
reference vector. The PLL synthesizes the voltage angle θv ; this
is the angle of a rotational reference frame that is aligned with
vector Vs . At any sampling time, the correct angle is found if
either (10) or (11) is true.
(10)
Vq = 0.
(11)
The projection of the voltage vector on the q-axis of this
reference frame is used as the error signal in order to enforce (11). The components Vα , Vβ are transformed into the
rotational reference frame given by θv and −Vq is fed into a
proportional and integral (PI) controller. The PI output is the
frequency estimate and is used in the integration of θv .
An important feature of this PLL is that the accuracy of frequency estimation depends only on the quality of the reference
vector. In a sensorless controlled IM drive, the stator voltage is
the very clean vector available (compared to current, flux, etc.),
especially if stator voltages are constructed using the switching
states of the inverter, noise, and offsets kept to a minimum.
Additionally, the voltage vector has a considerable magnitude
and produces a consistent large-enough error signal at the input
of the PI block in Fig. 3.
In the previously published work [1]–[3], the synchronous
frequency was determined based on IM (12) as
IV. D ESCRIPTION OF THE P HASE L OCKED L OOP
A typical three-phase UPS system uses two power feeders.
Input voltages of the first feeder are rectified and feed the dc
bus of an inverter. The second feeder (also called bypass) is
used as backup and is supposed to take over the loads if any
fault develops in the power conditioning circuit. This brings the
problem of synchronizing the output voltages of the inverter
with those of the bypass. Synchronization is required only if the
frequency of the bypass is within acceptable limits (59–61 Hz).
A PLL can be used to slowly lock the inverter output voltage
vector to the bypass vector from any initial condition. The
stability and dynamics of the synchronization are relatively easy
to achieve because the bypass voltage has an almost constant
magnitude and its frequency varies slowly in the 60-Hz vicinity.
Conditions are much tougher for a PLL intended for a
frequency estimation in an induction machine (IM) drive. This
is because the reference vector likely changes its frequency (and
its magnitude) much faster during transients. Additionally, an
estimation for a wider range and for both positive and negative
Vd = Vs ωs =
(Vβ − Rs Iβ )λα − (Vα − Rs Iα )λβ
.
λ2α + λ2β
(12)
The offsets and distortions in the fluxes and back EMFs, the
dependence on stator resistance, and the decreasing magnitude
of the numerator at low speeds deteriorate the ωs estimate.
V. C ONTROL S YSTEM D ESCRIPTION
The overall structure of the control system used in this paper
is presented in Fig. 4. The system uses a rotor direct field orientation control. Both a fixed-cutoff LPF and a programmable
LPF have been used for stator flux estimation and their outputs
are compared.
The angle of the rotor flux is computed using
θr = tan−1
λβ,r
.
λα,r
(13)
COMANESCU AND XU: AN IMPROVED FLUX OBSERVER FOR SENSORLESS VECTOR CONTROL OF INDUCTION MOTORS
53
Fig. 5. Real and estimated voltage angle at startup.
Fig. 4.
Structure of the control system.
The rotor speed is estimated as
1
Lm Iq
ωr =
·
ωs −
.
np
Tr |λr |
(14)
The stator voltages (line-neutral) are obtained using the duty
cycle of the inverter switches as
Vdc
(2Ta − Tb − Tc )
3
Vdc
(−Ta + 2Tb − Tc ).
=
3
Van =
(15)
Vbn
(16)
The stator voltages in the stationary reference frame are
Vα = Van
(17)
1
Vβ = √ (Van + 2Vbn ).
3
(18)
The system uses two stator flux estimators that run in parallel
and one rotor flux calculator. To focus on evaluating the PLLbased flux estimation, there is no active flux control loop
included.
VI. S IMULATION R ESULTS
The proposed control system has been simulated using
Simulink. A sensorless IM drive simulation was set up. The
motor is started using a traditional VM observer. The speed
reference is set at 500 r/min and at t = 0.3 s, and the step
changed to 1000 r/min. The load torque is 0.2 pu.
The dynamic performance of the PLL and the accuracy of the
frequency estimation process are analyzed. The PLL is operated
in parallel with the rest of the control system, and the stator
frequency angle θv and voltage Vq are monitored.
Fig. 5 shows the real angle of the stator voltage vector and
the estimated angle. In about 80 ms from startup, the angle
estimated by the PLL has converged to the real angle. The speed
reference change at t = 0.3 s produces very little disturbance in
the angle estimation.
Fig. 6. Estimated frequency and Vq at startup.
Fig. 6 shows the frequency estimate by both the PLL and
the classical method [using (12)]. The projection of the voltage
vector on the q-axis of the PLL reference frame can also
be seen.
In the simulation, the model adds noise and offsets (0.5% pu)
to the measured voltages/currents in order to account for the
signal acquisition hardware. In a real system, the projections
of the voltage vector have superimposed ac components; this
causes the estimation ripple in Fig. 6. At the end of the transient,
the estimated frequency has stabilized, and the PLL holds Vq
close to zero. When the step change at t = 0.3 s applies, the
frequency estimate locks to the new value very quickly. On
the other hand, it can be seen that the frequency estimated
by the traditional method has more ripples than that by the
PLL method (this is because the offsets in the current channels
intervene). Also, implementation of (12) produces some steadystate error even in simulation; estimation only gets worse if the
back EMFs or fluxes are distorted. The high-frequency burst
at the start of the simulation comes from the aggressive tuning
needed for the PI controller in Fig. 3.
Note that the final goal of using PLL is for the flux-estimation
improvement. Fig. 7 compares the fluxes estimated by the PLLbased algorithm to the real motor fluxes.
As evidenced by the results, the estimated fluxes are very
close to the real fluxes. The PLL-based programmable LPF
successfully overcomes transient of the speed/frequency at 0.3 s
and the estimated flux waveforms track the motor real fluxes
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 1, FEBRUARY 2006
Fig. 8.
Command and estimated frequency at 9–27–9-Hz step and 50 ms/div.
Fig. 9.
Command and estimated frequency at 9–45–9-Hz step and 50 ms/div.
Fig. 7. Real and estimated rotor fluxes.
TABLE I
MOTOR SPECIFICATIONS AND PARAMETERS
very well. It is also noticeable that for the initial several cycles,
however, the flux estimates are not quite right. This can be
attributed to the initial converging time needed by the PLL. It
can be suspected that this estimator might not be able to provide
flux information for the initial motor startup in the sensorless
vector control mode.
VII. E XPERIMENTAL R ESULTS
The induction motor used in the experimental testing is a
typical three-phase squirrel cage machine. The specifications
and parameters are listed in Table I.
The motor is powered by an insulated gate bipolar transistor
(IGBT) inverter. The DSP for implementing the controller
is a low cost, 16-bit fixed-point TMS320-F2407PGEA. The
software of the controller is organized in two interrupts. A fast
interrupt (50 µs) processes feedback signals, runs the classical
VM observer and the PLL-based programmable LPF, computes
the rotor flux angle and regulates the currents through two PI
controllers. A slow interrupt (100 µs) estimates and regulates
the rotor speed, executes the PLL algorithm and outputs the
PWM commands. The inverter switching frequency is 10 kHz.
The phase currents are measured through Hall sensors. Stator
voltages are computed using (15)–(18) with a dc bus voltage
sensor.
The initial investigation is focused on the dynamic performance of the PLL frequency estimator. To study that, a simple
V/f algorithm was used to run the motor. A step change in
frequency is applied and the dynamics of the PLL output is
recorded. Fig. 8 shows the command and estimated frequency
for a 9–27–9-Hz step changes.
Fig. 9 shows the same waveforms for a 9–45–9-Hz change.
Both tests are done at no load. It can be seen that, with proper
tuning, the PLL can estimate the stator frequency with high
bandwidth. The waveforms of fcommand in both Figs. 8 and 9
should be noise-free step functions, however, all signals are
shown noisy due to D/A and probe noise.
The second investigation is aimed at the stator flux waveforms produced by the PLL programmable LPF. The results
are compared with those estimated by the classical observer.
Both stator flux estimators run in parallel but only one rotor
flux calculator is used due to DSP time constraints.
At a convenient stator frequency, the system is transitioned
and field orientation is supported by the PLL-based programmable LPF. For comparison, the classical VM observer is
implemented at the same time with a cutoff frequency of
19.98 rad/s (3.18 Hz); this was selected experimentally. Frequencies lower than 3.18 Hz were tried, but the obtained flux
corresponding to (3) had significant offsets.
As discussed in the theory section, the PLL programmable
LPF is expected to produce stator fluxes very close to those of
the fixed-cutoff LPF for frequencies greater than 5ωc (17 Hz).
For lower frequencies, the classical observer’s output is lagging
the back EMFs by an angle less than 90◦ and leading the stator
flux obtained by the programmable LPF.
Fig. 10 shows the α-axis signals obtained by the proposed method. The test is done at no load. The back EMF,
COMANESCU AND XU: AN IMPROVED FLUX OBSERVER FOR SENSORLESS VECTOR CONTROL OF INDUCTION MOTORS
55
Fig. 10. Back EMF and stator flux by programmable LPF at f = 18 Hz,
k = 1, and 20 ms/div.
Fig. 12. Stator flux estimation from VM to PLL programmable LPF during
transition at 200 ms/div.
Fig. 11. Back EMF and stator fluxes by VM observer and PLL programmable
LPF at f = 2.1 Hz and k = 1, 0.2 s/div.
intermediate flux λ1αs , and the stator flux λαs are shown. The
stator frequency is about 18 Hz, and k = 1. For k = 1, the stator
flux λαs should√lag λ1αs by 45◦ and its magnitude should be
approximately 2 times bigger.
To verify the behavior of the PLL-based flux observer,
Fig. 11 was obtained at a low frequency (approx. 2.1 Hz) where
the difference between the two estimators is significant. The
back EMF is very small and noisy. The output of the VM
observer is incorrect in both magnitude and phase. The flux
given by the PLL-based programmable LPF lags the back EMF
by 90◦ .
Fig. 12 shows the stator flux estimate used as the input to
the rotor-flux-calculator block at the moment of transition. The
signal that commands the transition is on Channel (Ch.) 2.
On the left side of Fig. 12, the field orientation control was
done using the stator flux estimates produced by the classic
VM observer. After transition, the fluxes produced by the PLLbased programmable LPF are used. The operating frequency for
testing is approximately 8 Hz. Note the difference in magnitude
and phase angle between the two stator flux estimators.
The behavior in terms of offsets can also be observed in
Fig. 12. The fluxes given by the VM appear to have significant
offsets and the waveforms are shifted—this is due to the analog
channel and the A–D imbalance. Under the same circumstances, the programmable LPF yields flux waveforms that are
more symmetric because of the higher cutoff frequency used.
Fig. 13 shows the flux waveforms for a 100–700 r/min
step change in the speed command. The field orientation is
supported with the PLL-based estimator. The test is done at no
Fig. 13. Stator fluxes during 100–700 r/min step change at 100 ms/div.
load. It can be seen that the flux maintains its correct angle and
magnitude during the speed transient.
VIII. C ONCLUSION
This paper addresses the problems associated with the stator
flux estimation in a sensorless vector control of induction motor
drives. It is shown analytically and experimentally that the
traditional VM observer produces significant estimation errors
in the low-frequency range. If the stator frequency is known, a
programmable LPF and a vector rotator can be used to correct
the problem. Estimation of the stator frequency by a PLL is proposed and investigated. It is shown that a frequency estimation
can be achieved with a high bandwidth and robustly in transient.
Unlike the classical approach where frequency is estimated
analytically, the PLL relies only on the stator voltage vector
and is less influenced by offsets, distortions, or variations of the
stator resistance. It is shown experimentally that stator fluxes
obtained with the PLL-based estimator correct the magnitude
and the phase errors of the traditional VM observer and improve the performance of sensorless vector control of IMs. The
entire control algorithm is conveniently realized on a low-cost
DSP chip.
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 1, FEBRUARY 2006
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Mihai Comanescu (S’02) was born in Bucharest,
Romania, in 1968. He received the B.S. degree from
Bucharest Polytechnic Institute, Romania, in 1992
and the M.S. and Ph.D. degrees from The Ohio State
University, Columbus, in 2001 and 2005, respectively, all in electrical engineering.
He is currently with Azure Dynamics, Woburn,
MA, working on electric vehicle technology. His research interests include power electronics, ac drives,
and motion control systems.
Longya Xu (S’89–M’90–SM’93–F’04) received the
M.S. and Ph.D. degrees from the University of
Wisconsin, Madison, in 1986 and 1990, respectively,
all in electrical engineering.
He joined the Department of Electrical and
Computer Engineering at The Ohio State University,
Columbus, in 1990, where he is currently a Professor.
He has served as a Consultant to many industry
companies including Raytheon Company, U.S. Wind
Power Company, General Motors, Ford, and Unique
Mobility Inc. for various industrial concerns. His
research and teaching interests include dynamic modeling and optimized design
of electrical machines and power converters for variable speed generation
and drive systems, application of advanced control theory, and digital signal
processor for motion control and distributed power systems in super-high-speed
operation.
Dr. Xu received the 1990 First Prize Paper Award from the Industrial Drives
Committee, IEEE Industry Applications Society (IAS). In 1991, he won a
Research Initiation Award from the National Science Foundation. He is also
a recipient of the 1995 and 1999 Lumley Research Awards for his outstanding
research accomplishments from the College of Engineering, The Ohio State
University. He has served as the Chairman of the Electric Machine Committee
of the IAS and an Associate Editor of IEEE TRANSACTIONS ON POWER
ELECTRONICS.